Multidimensional free-energy calculations using the weighted histogram analysis method

The recently formulated weighted histogram analysis method (WHAM)¹ is an extension of Ferrenberg and Swendsen's multiple histogram technique for free-energy and potential of mean force calculations. As an illustration of the method, we have calculated the two-dimensional potential of mean force surface of the dihedrals gamma and chi in deoxyadenosine with Monte Carlo simulations using the all-atom and united-atom representation of the AMBER force fields. This also demonstrates one of the major advantages of WHAM over umbrella sampling techniques. The method also provides an analysis of the statistical accuracy of the potential of mean force as well as a guide to the most efficient use of additional simulations to minimize errors. © 1995 John Wiley & Sons, Inc.     

Absolute free-energy calculations of liquids using a harmonic reference state

Absolute free-energy methods provide a potential solution to the overlap problem in free-energy calculations. In this paper, we report an extension of the previously published confinement method (J. Phys. Chem. B 2006, 110, 17212-20) to fluid simulations. Absolute free energies of liquid argon and liquid water are obtained accurately and compared with results from thermodynamic integration. The method works by transforming the liquid state into a harmonic, solid reference state. This is achieved using a special restraint potential that allows molecules to change their restraint position during the simulation, which circumvents the need for the molecules to sample the full extent of their translational freedom. The absolute free energy of the completely restrained reference state is obtained from a normal mode calculation. Because of the generic reference state used, the method is applicable to nonhomogeneous, diffusive systems and could provide an alternative method in situations in which solute annihilation fails due to the size of the solute. Potential applications include calculation of solvation energies of large molecules and free energies of peptide conformational changes in explicit solvent.     

Free energy calculation methods: A theoretical and empirical comparison of numerical errors and a new method qualitative estimates of free energy changes

We present a comparison of four free energy calculation methods: thermodynamic integration (TI); traditional free energy perturbation (FEP); Bennett's acceptance ratio method (IPS); and a method that is related to an implementation of the WHAM method (CRS). The theoretical bases of the methods are first described, then calculations of the solvation free energies of methane and ethane are performed to determine the magnitude of the errors for the different methods. We find that the methods give similar errors when many intermediate states (windows) are used, but the IPS and CRS methods give smaller errors than the TI and FEP methods when no intermediate states are used. We also present a new procedure (based on the CRS method) that uses coordinates from simulations of a set of solutes to calculate the salvation free energies of additional solutes for which no simulations were performed. Solvation free energies for nine solutes (methanol, dimethylether, methylamine, methylammonium, dimethylamine, fluoromethane, difluoromethane, trifluoromethane, and tetrafluoromethane) are estimated based only on simulations of set of small hydrophobic solutes (including methane, ethane, and propane). These estimates can be surprisingly accurate and appear to be useful for making rapid estimates of solvation free energies. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 902–919, 1997     

Analysis of the statistical error in umbrella sampling simulations by umbrella integration

Umbrella sampling simulations, or biased molecular dynamics, can be used to calculate the free-energy change of a chemical reaction. We investigate the sources of different sampling errors and derive approximate expressions for the statistical errors when using harmonic restraints and umbrella integration analysis. This leads to generally applicable rules for the choice of the bias potential and the sampling parameters. Numerical results for simulations on an analytical model potential are presented for validation. While the derivations are based on umbrella integration analysis, the final error estimate is evaluated from the raw simulation data, and it may therefore be generally applicable as indicated by tests using the weighted histogram analysis method.     

Calculation of free-energy differences and potentials of mean force by a multi-energy gap method

A method is proposed to significantly accelerate the convergence of free-energy calculations. It introduces a bias factor in Monte Carlo simulations or, equivalently, a bias force in molecular dynamics simulations. The bias factor targets the energy gap, i.e., the difference in energy function between two states, and is therefore specifically designed for calculating free-energy differences. The goal is to make the probability density of the energy gap as uniform as possible, thus allowing for its accurate determination. An iterative procedure, based on simulations at higher temperatures, is devised to obtain the bias factor. The same method naturally extends to the calculation of potentials of mean force. The generalized coordinate, for which the potential of mean force is to be calculated, now plays the role of the energy gap. Applications to model systems confirm the expected increase in accuracy of calculated free-energy differences and potentials of mean force.     

Computing accurate potentials of mean force in electrolyte solutions with the generalized gradient-augmented harmonic Fourier beads method

We establish the accuracy of the novel generalized gradient-augmented harmonic Fourier beads (ggaHFB) method in computing free-energy profiles or potentials of mean force (PMFs) through comparison with two independent conventional techniques. In particular, we employ umbrella sampling with one dimensional weighted histogram analysis method (WHAM) and free molecular dynamics simulation of radial distribution functions to compute the PMF for the Na(+)-Cl(-) ion-pair separation to 16 A in 1.0M NaCl solution in water. The corresponding ggaHFB free-energy profile in six dimensional Cartesian space is in excellent agreement with the conventional benchmarks. We then explore changes in the PMF in response to lowering the NaCl concentration to physiological 0.3 and 0.1M, and dilute 0.0M concentrations. Finally, to expand the scope of the ggaHFB method, we formally develop the free-energy gradient approximation in arbitrary nonlinear coordinates. This formal development underscores the importance of the logarithmic Jacobian correction to reconstruct true PMFs from umbrella sampling simulations with either WHAM or ggaHFB techniques when nonlinear coordinate restraints are used with Cartesian propagators. The ability to employ nonlinear coordinates and high accuracy of the computed free-energy profiles further advocate the use of the ggaHFB method in studies of rare events in complex systems.     

Communication: multicanonical entropy-like solution of statistical temperature weighted histogram analysis method

A multicanonical update relation for calculation of the microcanonical entropy S(micro)(E) by means of the estimates of the inverse statistical temperature β(S), is proposed. This inverse temperature is obtained from the recently proposed statistical temperature weighted histogram analysis method (ST-WHAM). The performance of ST-WHAM concerning the computation of S(micro)(E) from canonical measures, in a model with strong free-energy barriers, is also discussed on the basis of comparison with the multicanonical simulation estimates.     

Ionic solvation studied by image-charge reaction field method

In a preceding paper [J. Chem. Phys. 131, 154103 (2009)], we introduced a new, hybrid explicit/implicit method to treat electrostatic interactions in computer simulations, and tested its performance for liquid water. In this paper, we report further tests of this method, termed the image-charge solvation model (ICSM), in simulations of ions solvated in water. We find that our model can faithfully reproduce known solvation properties of sodium and chloride ions. The charging free energy of a single sodium ion is in excellent agreement with the estimates by other electrostatics methods, while offering much lower finite-size errors. Similarly, the potentials of mean force computed for Na-Cl, Na-Na, and Cl-Cl pairs closely reproduce those reported previously. Collectively, our results demonstrate the superior accuracy of the proposed ICSM method for simulations of mixed media.     

Umbrella integration with higher-order correction terms

Umbrella integration is a method to analyze umbrella sampling simulations. It calculates free-energy changes from distributions obtained from molecular dynamics. While it can be formulated on the full sampled distributions, they are generally approximated by normal distributions. This is equivalent to the truncation of a power series of the free energy with respect to the reaction coordinate after the quadratic term or by a truncation of a cumulant expansion. Here, expressions for additional terms in the power series are derived. They can be calculated from the central moments of the distributions. This extension allows to test the approximations in applications.     

Accurate calculations of free-energy differences by the distribution method

We employ the strategy used in the successive umbrella sampling method [P. Virnau and M. Muller, J. Chem. Phys. 120, 10925 (2004)] to obtain the energy-difference distribution over its desired range. This is very helpful in calculating free-energy differences, where the source of the error is well recognized as the insufficient sampling over the relevant tail region in the energy-difference distribution. The distribution method proposed here employs the idea of restricting the sampling within an appropriate energy range, as was presented by Shing and Gubbins in their restricted umbrella sampling method [Mol. Phys. 46, 1109 (1982)]. We demonstrate the efficiency of the distribution method by calculating the free-energy difference of a model of harmonic oscillators where the systems exhibit nonoverlap features in their important phase spaces through the original Metropolis sampling. For this particular case, we show that the distribution method outperforms the free-energy perturbation method and even the Bennett's acceptance ratio method [J. Comput. Phys. 22, 245 (1976)] with the fastest convergence and the smallest relative errors. We further demonstrate the application of the distribution method with a simple point charge water model.     

Modeling the interplay between geometrical and energetic effects in protein folding

A theoretical framework is constructed with the aid of a free-energy functional method that is capable of describing the interplay between geometrical and energetic effects on protein folding. In this paper, we generalize a free-energy functional model based on polymer theory to make it more appropriate for comparison with protein folding simulations and experiments. This generalization is made by introducing cooperativity into the configurational entropy and the internal energy. Modifications to configurational entropy enable the model to account for the loop-loop interactions, a contribution neglected in the original model. Modifications to the internal energy introduce many-body corrections, which are needed to establish quantitative contact to simulations as well as experimental observations. To demonstrate the efficiency of the modified analytical model, we compare our results with C(alpha) structure-based (Go) model simulations of chymotrypsin inhibitor II and the SH3 domain of src.     

Modification and optimization of the united-residue (UNRES) potential energy function for canonical simulations. I. Temperature dependence of the effective energy function and tests of the optimization method with single training proteins

We report the modification and parametrization of the united-residue (UNRES) force field for energy-based protein structure prediction and protein folding simulations. We tested the approach on three training proteins separately: 1E0L (beta), 1GAB (alpha), and 1E0G (alpha + beta). Heretofore, the UNRES force field had been designed and parametrized to locate native-like structures of proteins as global minima of their effective potential energy surfaces, which largely neglected the conformational entropy because decoys composed of only lowest-energy conformations were used to optimize the force field. Recently, we developed a mesoscopic dynamics procedure for UNRES and applied it with success to simulate protein folding pathways. However, the force field turned out to be largely biased toward -helical structures in canonical simulations because the conformational entropy had been neglected in the parametrization. We applied the hierarchical optimization method, developed in our earlier work, to optimize the force field; in this method, the conformational space of a training protein is divided into levels, each corresponding to a certain degree of native-likeness. The levels are ordered according to increasing native-likeness; level 0 corresponds to structures with no native-like elements, and the highest level corresponds to the fully native-like structures. The aim of optimization is to achieve the order of the free energies of levels, decreasing as their native-likeness increases. The procedure is iterative, and decoys of the training protein(s) generated with the energy function parameters of the preceding iteration are used to optimize the force field in a current iteration. We applied the multiplexing replica-exchange molecular dynamics (MREMD) method, recently implemented in UNRES, to generate decoys; with this modification, conformational entropy is taken into account. Moreover, we optimized the free-energy gaps between levels at temperatures corresponding to a predominance of folded or unfolded structures, as well as to structures at the putative folding-transition temperature, changing the sign of the gaps at the transition temperature. This enabled us to obtain force fields characterized by a single peak in the heat capacity at the transition temperature. Furthermore, we introduced temperature dependence to the UNRES force field; this is consistent with the fact that it is a free-energy and not a potential energy function. beta     

Enhanced free-energy calculation using multiscale simulation

We propose a multiscale simulation method combining the efficiency of a coarse-grained model (CGM) and the accuracy of an all-atom model (AAM) for free-energy landscape calculation of protein systems. A protein's conformation space is quickly searched first using CGM. Then the obtained information is incorporated into AAM simulations. The free-energy landscape is subsequently obtained from AAM simulations. This method was tested on chignolin folding. The results demonstrated that the computational time was reduced by as much as 90%.     

Convergence and error estimation in free energy calculations using the weighted histogram analysis method

The weighted histogram analysis method (WHAM) has become the standard technique for the analysis of umbrella sampling simulations. In this article, we address the challenges (1) of obtaining fast and accurate solutions of the coupled nonlinear WHAM equations, (2) of quantifying the statistical errors of the resulting free energies, (3) of diagnosing possible systematic errors, and (4) of optimally allocating of the computational resources. Traditionally, the WHAM equations are solved by a fixed-point direct iteration method, despite poor convergence and possible numerical inaccuracies in the solutions. Here, we instead solve the mathematically equivalent problem of maximizing a target likelihood function, by using superlinear numerical optimization algorithms with a significantly faster convergence rate. To estimate the statistical errors in one-dimensional free energy profiles obtained from WHAM, we note that for densely spaced umbrella windows with harmonic biasing potentials, the WHAM free energy profile can be approximated by a coarse-grained free energy obtained by integrating the mean restraining forces. The statistical errors of the coarse-grained free energies can be estimated straightforwardly and then used for the WHAM results. A generalization to multidimensional WHAM is described. We also propose two simple statistical criteria to test the consistency between the histograms of adjacent umbrella windows, which help identify inadequate sampling and hysteresis in the degrees of freedom orthogonal to the reaction coordinate. Together, the estimates of the statistical errors and the diagnostics of inconsistencies in the potentials of mean force provide a basis for the efficient allocation of computational resources in free energy simulations.     

Improved precision and efficiency of free energy calculations for small systems using lambda-scaled atomic masses and separating conformational and transformational sampling

We present results showing the importance of appropriate treatment of atomic masses in molecular dynamics (MD)-based single topology free-energy perturbations (FEPs) on small molecule systems. The reversibility of gas phase simulations is significantly improved by scaling the atomic mass of mutated atoms with the lambda variable normally used for the scaling of energy terms. Because this effect is less pronounced for solvated systems, it will not cancel in estimates of the relative hydration free energy difference. The advantage of mass scaling is demonstrated by a null mutation of ethane to ethane and the calculation of the relative hydration free energy difference between ethane and n-propane. Furthermore, it is found that the simulation time necessary for converged MD/FEPs is prohibitively large for relative hydration free energy calculations on cyclic alkanes. Therefore, we explore an alternative free energy pathway including strongly constrained conformations to improve convergence in FEP simulations of flexible molecules.     

Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials

A novel test-area (TA) technique for the direct simulation of the interfacial tension of systems interacting through arbitrary intermolecular potentials is presented in this paper. The most commonly used method invokes the mechanical relation for the interfacial tension in terms of the tangential and normal components of the pressure tensor relative to the interface (the relation of Kirkwood and Buff [J. Chem. Phys. 17, 338 (1949)]). For particles interacting through discontinuous intermolecular potentials (e.g., hard-core fluids) this involves the determination of delta functions which are impractical to evaluate, particularly in the case of nonspherical molecules. By contrast we employ a thermodynamic route to determine the surface tension from a free-energy perturbation due to a test change in the surface area. There are important distinctions between our test-area approach and the computation of a free-energy difference of two (or more) systems with different interfacial areas (the method of Bennett [J. Comput. Phys. 22, 245 (1976)]), which can also be used to determine the surface tension. In order to demonstrate the adequacy of the method, the surface tension computed from test-area Monte Carlo (TAMC) simulations are compared with the data obtained with other techniques (e.g., mechanical and free-energy differences) for the vapor-liquid interface of Lennard-Jones and square-well fluids; the latter corresponds to a discontinuous potential which is difficult to treat with standard methods. Our thermodynamic test-area approach offers advantages over existing techniques of computational efficiency, ease of implementation, and generality. The TA method can easily be implemented within either Monte Carlo (TAMC) or molecular-dynamics (TAMD) algorithms for different types of interfaces (vapor-liquid, liquid-liquid, fluid-solid, etc.) of pure systems and mixtures consisting of complex polyatomic molecules.     

Improving the efficiency and reliability of free energy perturbation calculations using overlap sampling methods

A challenge in free energy calculation for complex molecular systems by computer simulation is to obtain a reliable estimate within feasible computational time. In this study, we suggest an answer to this challenge by exploring a simple method, overlap sampling (OS), for producing reliable free-energy results in an efficient way. The formalism of the OS method is based on ensuring sampling of important overlapping phase space during perturbation calculations. This technique samples both forward and reverse free energy perturbation (FEP) to improve the free-energy calculation. It considers the asymmetry of the FEP calculation and features an ability to optimize both the precision and the accuracy of the measurement without affecting the simulation process itself. The OS method is tested at two optimization levels: no optimization (simple OS), and full optimization (equivalent to Bennett's method), and compared to conventional FEP techniques, including the widely used direct FEP averaging method, on three alchemical mutation systems: (a) an anion transformation in water solution, (b) mutation between methanol and ethane, and (c) alchemical change of an adenosine molecule. It is consistently shown that the reliability of free-energy estimates can be greatly improved using the OS techniques at both optimization levels, while the performance of Bennett's method is particularly striking. In addition, the efficiency of a calculation can be significantly improved because the method is able to (a) converge to the right answer quickly, and (b) work for large perturbations. The basic two-stage OS method can be extended to admit additional stages, if needed. We suggest that the OS method can be used as a general perturbation technique for computing free energy differences in molecular simulations.     

Application of computer simulation free-energy methods to compute the free energy of micellization as a function of micelle composition. 1. Theory

The widespread use of surfactant mixtures and surfactant/solubilizate mixtures in practical applications motivates the development of predictive theoretical approaches to improve fundamental understanding of the behavior of these complex self-assembling systems and to facilitate the design and optimization of new surfactant and surfactant/solubilizate mixtures. This paper is the first of two articles introducing a new computer simulation-free-energy/molecular thermodynamic (CS-FE/MT) model. The two articles explore the application of computer simulation free-energy methods to quantify the thermodynamics associated with mixed surfactant/cosurfactant and surfactant/solubilizate micelle formation in aqueous solution. In this paper (article 1 of the series), a theoretical approach is introduced to use computer simulation free-energy methods to compute the free-energy change associated with changing micelle composition (referred to as DeltaDeltaGi). In this approach, experimental critical micelle concentration (CMC) data, or a molecular thermodynamic model of micelle formation, is first used to evaluate the free energy associated with single (pure) surfactant micelle formation, g(form,single), in which the single surfactant micelle contains only surfactant A molecules. An iterative approach is proposed to combine the estimated value of gform,single with free-energy estimates of DeltaDeltaGi based on computer simulation to determine the optimal free energy of mixed micelle formation, the optimal micelle aggregation number and composition, and the optimal bulk solution composition. After introducing the CS-FE/MT modeling framework, a variety of free-energy methods are briefly reviewed, and the selection of the thermodynamic integration free-energy method is justified and selected to implement the CS-FE/MT model. An alchemical free-energy pathway is proposed to allow evaluation of the free-energy change associated with exchanging a surfactant A molecule with a surfactant/solubilizate B molecule through thermodynamic integration. In article 2 of this series, the implementation of the CS-FE/MT model to make DeltaDeltaGi free-energy predictions for several surfactant/solubilizate systems is discussed, and the predictions of the CS-FE/MT model are compared with the DeltaDeltaGi predictions of a molecular thermodynamic model fitted to relevant experimental data.     

QM/MM-PBSA method to estimate free energies for reactions in proteins

We have developed a method to estimate free energies of reactions in proteins, called QM/MM-PBSA. It estimates the internal energy of the reactive site by quantum mechanical (QM) calculations, whereas bonded, electrostatic, and van der Waals interactions with the surrounding protein are calculated at the molecular mechanics (MM) level. The electrostatic part of the solvation energy of the reactant and the product is estimated by solving the Poisson-Boltzmann (PB) equation, and the nonpolar part of the solvation energy is estimated from the change in solvent-accessible surface area (SA). Finally, the change in entropy is estimated from the vibrational frequencies. We test this method for five proton-transfer reactions in the active sites of [Ni,Fe] hydrogenase and copper nitrite reductase. We show that QM/MM-PBSA reproduces the results of a strict QM/MM free-energy perturbation method with a mean absolute deviation (MAD) of 8-10 kJ/mol if snapshots from molecular dynamics simulations are used and 4-14 kJ/mol if a single QM/MM structure is used. This is appreciably better than the original QM/MM results or if the QM energies are supplemented with a point-charge model, a self-consistent reaction field, or a PB model of the protein and the solvent, which give MADs of 22-36 kJ/mol for the same test set.